Chapter 8, Problem 20RE

To determine

To find:The remaining angle(s) and side(s) of the triangle, if it(they) exist.

Answer to Problem 20RE

No triangleexists with the given measurements.

Explanation of Solution

Given:

The sides $a=3$ , $c=1$ and angle $C=20°$ .

Calculation:

The Sine Law for a triangle $\Delta ABC$ is given by $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$ , where capital letters represent the angles and the small letters represent the length of the respective sides of the triangle.

Given the sides a andc , and the angle C , angle A can be determined as follows:

  $\begin{array}{c}\frac{\sin A}{a}=\frac{\sin C}{c}\\ \frac{\sin A}{3}=\frac{\sin 20°}{1}\\ \sin A=3\sin 20°\\ =1.026\end{array}$

Since the range of sine is $\left[-1,1\right]$ , $\sin A\ne 1.026$ .

So, no triangle exists with the given measurements.